Geodesic
Exact bounds on the closeness between the Student and standard normal distributions
Pinelis, Iosif1 
1 Department of Mathematical Sciences, Michigan Technological University, Houghton, 49931 Michigan, USA
ESAIM: Probability and Statistics, Tome 19 (2015), pp. 24-27.

Voir la notice de l'article provenant de la source Numdam

Upper bounds on the Kolmogorov distance (and, equivalently in this case, on the total variation distance) between the Student distribution with p degrees of freedom (SD p ) and the standard normal distribution are obtained. These bounds are in a certain sense best possible, and the corresponding relative errors are small even for moderate values of p. The same bounds hold on the closeness between SD p and SD q with q>p. Comparisons with known bounds are made.

Reçu le :
DOI : 10.1051/ps/2014014
Classification : 62E17, 60E15, 62E20, 62E15
Keywords: Student’s distribution, standard normal distribution, Kolmogorov distance, total variation distance, probability inequalities

Pinelis, Iosif 1

1 Department of Mathematical Sciences, Michigan Technological University, Houghton, 49931 Michigan, USA 
@article{PS_2015__19__24_0,
     author = {Pinelis, Iosif},
     title = {Exact bounds on the closeness between the {Student} and standard normal distributions},
     journal = {ESAIM: Probability and Statistics},
     pages = {24--27},
     publisher = {EDP-Sciences},
     volume = {19},
     year = {2015},
     doi = {10.1051/ps/2014014},
     mrnumber = {3374867},
     zbl = {1332.62054},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/ps/2014014/}
}
TY  - JOUR
AU  - Pinelis, Iosif
TI  - Exact bounds on the closeness between the Student and standard normal distributions
JO  - ESAIM: Probability and Statistics
PY  - 2015
SP  - 24
EP  - 27
VL  - 19
PB  - EDP-Sciences
UR  - http://geodesic.mathdoc.fr/articles/10.1051/ps/2014014/
DO  - 10.1051/ps/2014014
LA  - en
ID  - PS_2015__19__24_0
ER  - 
%0 Journal Article
%A Pinelis, Iosif
%T Exact bounds on the closeness between the Student and standard normal distributions
%J ESAIM: Probability and Statistics
%D 2015
%P 24-27
%V 19
%I EDP-Sciences
%U http://geodesic.mathdoc.fr/articles/10.1051/ps/2014014/
%R 10.1051/ps/2014014
%G en
%F PS_2015__19__24_0
Pinelis, Iosif. Exact bounds on the closeness between the Student and standard normal distributions. ESAIM: Probability and Statistics, Tome 19 (2015), pp. 24-27. doi : 10.1051/ps/2014014. http://geodesic.mathdoc.fr/articles/10.1051/ps/2014014/

S. Bourguin and C. Tudor, Malliavin calculus and self normalized sums. Seminaire de Propabilités XLV. Lect. Notes Math. Springer (2013) 323. | MR | Zbl

T. Cacoullos, V. Papathanasiou, and S.A. Utev, Variational inequalities with examples and an application to the central limit theorem. Ann. Probab. 22 (1994) 1607–1618. | MR | Zbl

I. Pinelis, Monotone tail and moment ratio properties of Student’s family of distributions. Version 2. (2012). | arXiv

I. Pinelis, On the Berry–Esseen bound for the Student statistic. Version 2. (2012). | arXiv

I. Pinelis, Exact bounds on the closeness between the Student and standard normal distributions. Version 2. (2013). | arXiv | MR

I. Pinelis and R. Molzon, Berry-Esséen bounds for general nonlinear statistics, with applications to Pearson’s and non-central Student’s and Hotelling’s. (2009). | arXiv

R.S. Pinkham and M.B. Wilk, Tail areas of the t-distribution from a Mills’-ratio-like expansion. Ann. Math. Statist. 34 (1963) 335–337. | MR | Zbl

H. Scheffé, A useful convergence theorem for probability distributions. Ann. Math. Statist. 18 (1947) 434–438. | MR | Zbl

R. Shimizu, Error bounds for asymptotic expansion of the scale mixtures of the normal distribution. Ann. Inst. Statist. Math. 39 (1987) 611–622. | MR | Zbl

Cité par Sources :